Quantum Algorithms for Tree Isomorphism and State Symmetrization

Abstract: The graph isomorphism problem is theoretically interesting and also has many practical applications. The best known classical algorithms for graph isomorphism all run in time super-polynomial in the size of the graph in the worst case. An interesting open problem is whether quantum computers can solve the graph isomorphism problem in polynomial time. In this paper, an algorithm is shown which can decide if two rooted trees are isomorphic in polynomial time. Although this problem is easy to solve efficiently on a classical computer, the techniques developed may be useful as a basis for quantum algorithms for deciding isomorphism of more interesting types of graphs. The related problem of quantum state symmetrization is also studied. A polynomial time algorithm for the problem of symmetrizing a set of orthonormal states over an arbitrary permutation group is shown.